Comments on: Singular Integral Operators and Convergence of Fourier Series http://cornellmath.wordpress.com/2008/02/12/singular-integral-operators-and-convergence-of-fourier-series/ Geometry, Topology, Categories, Groups, Physics, . . . Everything Sat, 19 Jul 2008 10:26:37 +0000 http://wordpress.org/?v=MU By: wlfdiver http://cornellmath.wordpress.com/2008/02/12/singular-integral-operators-and-convergence-of-fourier-series/#comment-2837 wlfdiver Mon, 25 Feb 2008 00:15:15 +0000 http://cornellmath.wordpress.com/?p=229#comment-2837 Oh my God you have eroded part of my brain! Oh my God you have eroded part of my brain!

]]> By: Ars Mathematica » Blog Archive http://cornellmath.wordpress.com/2008/02/12/singular-integral-operators-and-convergence-of-fourier-series/#comment-2835 Ars Mathematica » Blog Archive Sun, 24 Feb 2008 05:19:51 +0000 http://cornellmath.wordpress.com/?p=229#comment-2835 [...] intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the [...] [...] intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the [...]

]]> By: Peter Luthy http://cornellmath.wordpress.com/2008/02/12/singular-integral-operators-and-convergence-of-fourier-series/#comment-2807 Peter Luthy Tue, 12 Feb 2008 23:07:11 +0000 http://cornellmath.wordpress.com/?p=229#comment-2807 I'm glad that you enjoyed it, Terry. Thanks for the link to Stein's paper - I had never actually read this myself, though I am familiar with the result. As is usually the case with what he produces, it's remarkably well-written. Khinchin's (I always use the shortest romanization since there are so many) inequality is a great trick to have around. We used it several times in a course I took from my adviser (Camil Muscalu) last term. As far as the intuition of pointwise convergence and boundedness of maximal operators being equivalent problems is concerned, I've been reading your paper on the interplay between maximal operators and ergodic theory which certainly provides additional motivation for believing such a claim: http://arxiv.org/abs/math/0510581 I’m glad that you enjoyed it, Terry. Thanks for the link to Stein’s paper - I had never actually read this myself, though I am familiar with the result. As is usually the case with what he produces, it’s remarkably well-written.

Khinchin’s (I always use the shortest romanization since there are so many) inequality is a great trick to have around. We used it several times in a course I took from my adviser (Camil Muscalu) last term. As far as the intuition of pointwise convergence and boundedness of maximal operators being equivalent problems is concerned, I’ve been reading your paper on the interplay between maximal operators and ergodic theory which certainly provides additional motivation for believing such a claim:

http://arxiv.org/abs/math/0510581

]]> By: Terence Tao http://cornellmath.wordpress.com/2008/02/12/singular-integral-operators-and-convergence-of-fourier-series/#comment-2803 Terence Tao Tue, 12 Feb 2008 19:43:57 +0000 http://cornellmath.wordpress.com/?p=229#comment-2803 Nice post! I just wanted to comment though that there is also a (partial) reverse implication: if a.e. pointwise convergence for Fourier summation was true in $latex L^p$ for some $latex 1 \leq p \leq 2$, then there must be a weak $latex (p,p)$ inequality for the corresponding maximal operator. This is known as Stein's maximum principle and is proven in this 1960 paper of Stein: http://www.jstor.org/view/0003486x/di961782/96p0007b/0 It relies primarily on the translation invariance of the situation and on Khintchine's inequality for random signs. Unfortunately the principle is known to fail for some operators when $latex p>2$ but it is still good intuition to think of pointwise convergence and maximal operator bounds as being morally equivalent. For the easier question of norm convergence, we have the uniform boundedness principle, which shows that if we have $latex T_n f$ converging in $latex L^p$ to f for all f in $latex L^p$, then the $latex T_n$ must be uniformly bounded in the $latex L^p$ operator norm. Of course, it is easier to bound each of the $latex T_n$ individually than it is to bound the maximal operator $latex \sup_n T_n$, which is why pointwise convergence questions tend to be significantly harder than norm convergence questions. (The dominated convergence theorem also gives another indication why this should be the case.) Nice post! I just wanted to comment though that there is also a (partial) reverse implication: if a.e. pointwise convergence for Fourier summation was true in L^p for some 1 \leq p \leq 2, then there must be a weak (p,p) inequality for the corresponding maximal operator. This is known as Stein’s maximum principle and is proven in this 1960 paper of Stein:

http://www.jstor.org/view/0003486x/di961782/96p0007b/0

It relies primarily on the translation invariance of the situation and on Khintchine’s inequality for random signs. Unfortunately the principle is known to fail for some operators when p>2 but it is still good intuition to think of pointwise convergence and maximal operator bounds as being morally equivalent.

For the easier question of norm convergence, we have the uniform boundedness principle, which shows that if we have T_n f converging in L^p to f for all f in L^p, then the T_n must be uniformly bounded in the L^p operator norm. Of course, it is easier to bound each of the T_n individually than it is to bound the maximal operator \sup_n T_n, which is why pointwise convergence questions tend to be significantly harder than norm convergence questions. (The dominated convergence theorem also gives another indication why this should be the case.)

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